We can define a subclass of the regular languages. Fix an alphabet $\Sigma$. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field of DNA computing. AFAICT, that's a different class of languages).
A language $L$ is circular if and only if for all words $w \in \Sigma$, we have:
$w\in L$ if and only if, for all integers $k > 0$ we have $w^k\in L$.
Is this class of languages known? I am interested in:
a name for it
decidability of the problem, given an automaton (in particular: a DFA), whether the accepted language obeys to the above definition
a "nice" characterization (e.g. equational?) of the definition.